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<H1><A NAME="SECTION03400000000000000000"></A><A NAME="chapaccstab"></A>
<BR>
Accuracy and Stability
</H1>

<P>

<P>

<P>

<P>
In addition to providing faster routines than previously available,
LAPACK provides more comprehensive and better<A NAME="9688"></A>
error<A NAME="9689"></A> bounds<A NAME="9690"></A>.
Our goal is to provide error bounds for most quantities computed by LAPACK.

<P>
In this chapter we explain our overall approach to obtaining error bounds,
and provide enough information to use the software.
The comments at the beginning of the individual routines should be consulted
for more details.
It is beyond the scope of this chapter to justify all the bounds
we present. Instead, we give references to the literature.
For example, standard material on error analysis can be found
in [<A
 HREF="node151.html#demmelMA221">25</A>,<A
 HREF="node151.html#GVL2">55</A>,<A
 HREF="node151.html#higham96">67</A>,<A
 HREF="node151.html#stewartsun90">95</A>].

<P>
In order to make this chapter easy to read, we have labeled sections
not essential for a first reading as <B>Further Details</B>. The sections
not labeled as <B>Further Details</B> should provide all the information
needed to understand and use the main error bounds computed by
LAPACK.  The <B>Further Details</B> sections provide mathematical background,
references, and tighter but more expensive error bounds, and may be read later.

<P>
In section&nbsp;<A HREF="node73.html#secroundoff">4.1</A>
we discuss the sources of numerical error, in particular roundoff
error.
Section&nbsp;<A HREF="node75.html#secnormnotation">4.2</A> discusses how to measure errors,
as well as some standard notation.
Section&nbsp;<A HREF="node77.html#secwhereerrorboundscomefrom">4.3</A> discusses further details
of how error bounds are derived.
Sections&nbsp;<A HREF="node80.html#secAx_b">4.4</A> through <A HREF="node106.html#secGSVDbound">4.12</A> present error bounds
for linear equations, linear least squares problems,
generalized linear least squares problems,
the symmetric eigenproblem,
the nonsymmetric eigenproblem,
the singular value decomposition,
the generalized symmetric definite eigenproblem,
the generalized nonsymmetric eigenproblem
and the
generalized (or quotient) singular value decomposition respectively.
Section&nbsp;<A HREF="node108.html#secfastblas">4.13</A> discusses the impact of fast
Level 3 BLAS<A NAME="9701"></A> on
the accuracy<A NAME="9702"></A> of LAPACK routines.

<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>

<UL>
<LI><A NAME="tex2html5169"
 HREF="node73.html">Sources of Error in Numerical Calculations</A>
<UL>
<LI><A NAME="tex2html5170"
 HREF="node74.html">Further Details:  Floating Point Arithmetic</A>
</UL>
<LI><A NAME="tex2html5171"
 HREF="node75.html">How to Measure Errors</A>
<UL>
<LI><A NAME="tex2html5172"
 HREF="node76.html">Further Details:  How to Measure Errors</A>
</UL>
<LI><A NAME="tex2html5173"
 HREF="node77.html">Further Details:  How Error Bounds Are Derived</A>
<UL>
<LI><A NAME="tex2html5174"
 HREF="node78.html">Standard Error Analysis</A>
<LI><A NAME="tex2html5175"
 HREF="node79.html">Improved Error Bounds</A>
</UL>
<LI><A NAME="tex2html5176"
 HREF="node80.html">Error Bounds for Linear Equation Solving</A>
<UL>
<LI><A NAME="tex2html5177"
 HREF="node81.html">Further Details:  Error Bounds for Linear Equation Solving</A>
</UL>
<LI><A NAME="tex2html5178"
 HREF="node82.html">Error Bounds for Linear Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html5179"
 HREF="node83.html">Further Details:  Error Bounds for Linear Least Squares
Problems</A>
</UL>
<LI><A NAME="tex2html5180"
 HREF="node84.html">Error Bounds for Generalized Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html5181"
 HREF="node85.html">Linear Equality Constrained Least Squares Problem</A>
<UL>
<LI><A NAME="tex2html5182"
 HREF="node86.html">Further Details:
Error Bounds for Linear Equality Constrained Least Squares Problems</A>
</UL>
<LI><A NAME="tex2html5183"
 HREF="node87.html">General Linear Model Problem</A>
<UL>
<LI><A NAME="tex2html5184"
 HREF="node88.html">Further Details:  Error Bounds for General Linear Model Problems</A>
</UL>
</UL>
<LI><A NAME="tex2html5185"
 HREF="node89.html">Error Bounds for the Symmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html5186"
 HREF="node90.html">Further Details:  Error Bounds for the Symmetric Eigenproblem </A>
</UL>
<LI><A NAME="tex2html5187"
 HREF="node91.html">Error Bounds for the Nonsymmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html5188"
 HREF="node92.html">Further Details:  Error Bounds for the Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5189"
 HREF="node93.html">Overview</A>
<LI><A NAME="tex2html5190"
 HREF="node94.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5191"
 HREF="node95.html">Computing <B><I>s</I></B> and <IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.gif"
 ALT="${\rm sep}$"></A>
</UL>
</UL>
<LI><A NAME="tex2html5192"
 HREF="node96.html">Error Bounds for the Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html5193"
 HREF="node97.html">Further Details:  Error Bounds for the Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html5194"
 HREF="node98.html">Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5195"
 HREF="node99.html">Further Details:  Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
</UL>
<LI><A NAME="tex2html5196"
 HREF="node100.html">Error Bounds for the Generalized Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5197"
 HREF="node101.html">Further Details: Error Bounds for the Generalized
Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5198"
 HREF="node102.html">Overview</A>
<LI><A NAME="tex2html5199"
 HREF="node103.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5200"
 HREF="node104.html">Computing <B><I>s</I><SUB><I>i</I></SUB></B>, <IMG
 WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$l_{\cal I}$">,
<IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$r_{\cal I}$">
and
<IMG
 WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="${\rm Dif}_u$">,
<IMG
 WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="${\rm Dif}_l$"></A>
<LI><A NAME="tex2html5201"
 HREF="node105.html">Singular Eigenproblems</A>
</UL>
</UL>
<LI><A NAME="tex2html5202"
 HREF="node106.html">Error Bounds for the Generalized Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html5203"
 HREF="node107.html">Further Details:  Error Bounds for the Generalized Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html5204"
 HREF="node108.html">Error Bounds for Fast Level 3 BLAS</A>
</UL>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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